Israel is home to Jews of many different ethnic origins, including the Ashkenazim (with proximate roots in Central Europe), the Sephardim (with proximate roots in the Iberian peninsula), and the Mizrahim (with proximate roots in the Middle East).
The Chief Rabbinate of Israel is a spiritual authority that consists of two Chief Rabbis, one Ashkenazi and one Sephardi. The Rabbinate is part of the Israeli judicial system and therefore holds considerable power over personal life and status in Israel.
Why is there no Mizrahi Chief Rabbi, even though the Mizrahim outnumber the Sephardim two-to-one in Israel?
It’s a historical artifact, basically. The office of Sephardic Chief Rabbi was carried over from the equivalent office from Ottoman Empire rule and the Ottomans ruled over the Mizrahim for the most part, so (what is now called) the Sephardic Chief Rabbi was their top Rabbinic authority as well, despite their distinct customs and culture.
The Ashkenazic Chief Rabbi was added by the British during the Mandate period because the British wanted someone to deal with who could authoritatively represent the growing European immigrant Jewish community.
The State of Israel simply kept the existing pre-state offices.
I see. Have there been any notable movements to change this state of affairs by adding another Chief Rabbi? Do any major political parties or religious groups support such a reform?
The OP seems to think that Sephardim and Mizrachim ar two separate groups while in fact, there’s a vast overlap between the two. Sephardim are Jews descended, at least in part, from the Jews of medieval Spain, while Mizrachim are Jews originating from Muslim countries. Jews from North Africa, the Levant and Iraq are both Sephardim and Mizrachim; they also make up the majority of both groups in Israel. In fact, the only Mizrachim who aren’t also Sephardim are the Persian and Yemenite Jews, along with much smaller communities from Central Asia. On the other hand, the only major surviving non-Mizrachi Sehpardic communities are from the Balkans, and they’re not very religious anyway.
Every single Rishon Lezion (Sephardic Chief Rabbi) in Israeli history has been of Middle Eastern- Mizrachi - origin. The current one is Iraqi.
The other reason is that in this case, the distinction between Ashkenazim and Sephardim is liturgical. There is no “Mizrachi” siddur (prayer book), nor are there any halachot (religious rules) that apply specifically to Mizrachim. “Mizrachi” may be an ethnic classification, but it isn’t a religious one.